Loomis Type Theorem for Bounded Sequences

This paper extends Loomis’s classical spectral criterion for almost periodic functions to bounded sequences in discrete settings. We first establish a discrete Kadets-type theorem for bounded sequences by regarding almost periodic functions as continuous with respect to the Bohr metric. We then introduce three spectra for bounded sequences via the Carleman transform and prove their equivalence. Using the Beurling–Gelfand theorem, we derive a discrete Loomis-type theorem, providing a spectral criterion for almost periodicity of bounded sequences. Our results extend the continuous theory to the discrete case and offer new tools for analyzing almost periodicity in sequence spaces.